User blog:Allam948736/Allam's Extended Steinhaus-Moser Notation
Steinhaus-Moser notation can be essentially thought of as a fast iteration hierarchy where f_0(n) = n^n, as is stated on the article here. Based on this, I defined my own extension of Steinhaus-Moser notation, where n in a circle is f_w(n) in the aforementioned hierarchy, or n in an n+3-gon. I denote n in a circle as nw, because a circle can be thought of as a polygon with infinitely many (or omega) sides. Below are the first few values of nw. 1w = 14 = 1 2w = 25 (mega) 3w = 36 (grand megision) 4w = 47 5w = 58 To continue, I moved up to 3-dimensional shapes, and defined n in a tetrahedron (or n3) as n in n circles. In the hierarchy that Steinhaus-Moser notation forms, this would be f_w+1(n). 23 is comparable to and "slightly" greater than Moser's number (being equal to mega+ 3), and 643 is greater than Graham's number. I next defined n in a cube as n in n tetrahedrons (which would be f_w+2(n) in the hierarchy formed by Steinhaus-Moser notation). 23 is already much greater than Graham's number and even G(G(1)), and it is a number that I refer to as the cube mega, from the shape that the 2 would be in and the number mega, and used Aarex's naming scheme for numbers defined with the regular Steinhaus-Moser notation to name 33, 43, 53, etc. I then continued with dodecahedrons (which have pentagonal faces) and icosahedrons, and then defined n in a sphere (or n3) as n3. This has a growth rate of f_2w in the fast-growing hierarchy. I then continued with 4-dimensional shapes, 5-dimensional shapes, 6-dimensional shapes, etc. Although it is impossible to actually visualize these, they can at least be denoted as nk, where k is the number of dimensions. I then continued to an infinite number of dimensions, defining nw as nn+1, and nw (that would be n in an infinite-dimensional sphere, as crazy as that sounds) as nn+1 or nn+1. This achieves a growth rate of f_w^2 in the fast-growing hierarchy. But I didn't stop there. I next defined n2, 2 as w applied to n n times, n2, 2 as 2, 2 applied to n n times, and so on, until n2, 2 which is n2, 2, and continued with n3, 2 being 2, 2 applied to n n times. In general, each time you increase the first number in the array by 1, you apply the previous operator the input number of times (that is, nb, c is b, c applied to n n times if a > 3). This is true for any number of arguments. So let's skip to nw, 2, which is equal to nn+1, 2. In general, nw, b is equal to nn+1, b. At nw, 2 (which is nn+1, 2), we reach the end of the operators with 2 as the third argument in the array. This achieves the growth rate of f_(w^2)*2(n) in the fast-growing hierarchy, and to continue, I simply defined n2, 3 as w, 2 applied to n n times. Eventually, we reach nw, w, which has a growth rate of f_w^3(n) in the fast-growing hierarchy, and is equal to nw, n, which is nn+1, n, or nn+1, n We can just add more arguments to the array after that. In general, n2, 2, 2, ..., 2, 2, 2 w/ m 2s is equal to w, w, ..., w, w, w w/ m omegas applied to n n times, and nw, w,..., w, w, w w/ m omegas is equal to nw, w, ..., w, w, n w/ m-1 omegas which is in turn equal to nn+1, n, n, ..., n, n w/ m-2 copies of n. This achieves a growth rate of f_w^w in the fast-growing hierarchy, on par with Bowers' linear arrays. Category:Blog posts